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Nabla Dynamic Equations

  • Douglas Anderson
  • John Bullock
  • Lynn Erbe
  • Allan Peterson
  • HoaiNam Tran

Abstract

If \( \mathbb{T} \) has a right-scattered minimum m, define \( \mathbb{T}_\kappa : = \mathbb{T} - \{ m\} \) ; otherwise, set \( \mathbb{T}_\kappa = \mathbb{T} \) . The backwards graininess \( \nu :\mathbb{T}_\kappa \to \mathbb{R}_0^ + \) is defined by
$$ \nu (t) = t - \rho (t). $$
For \( f:\mathbb{T} \to \mathbb{R} \) and \( t \in \mathbb{T}_\kappa \) , define the nabla derivative [42] of f at t, denoted f (t), to be the number (provided it exists) with the property that given any ε > 0, there is a neighborhood U of t such that
$$ |f(\rho (t)) - f(s) - f^\nabla (t)(\rho (t) - s)| \leqslant \varepsilon |\rho (t) - s) $$
for all sU. For \( \mathbb{T} = \mathbb{R} \) , we have f =f′, the usual derivative, and for \( \mathbb{T} = \mathbb{Z} \) we have the backward difference operator, f (t)=∇f(t):=f(t)-f(t-1). Note that the nabla derivative is the alpha derivative when α = p. Many of the results in this chapter can be generalized to the alpha derivative case. Many of the results in this chapter can be found in [35, 37].

Keywords

Prove Theorem Adjoint Equation Semigroup Property Cauchy Function Constant Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Douglas Anderson
    • 1
  • John Bullock
    • 1
  • Lynn Erbe
    • 2
  • Allan Peterson
    • 2
  • HoaiNam Tran
    • 2
  1. 1.Department of Mathematics and Computer ScienceConcordia CollegeMoorheadUSA
  2. 2.Department of Mathematics and StatisticsUniversity of Nebraska-LincolnLincolnUSA

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